3.3: Solving Quadratic Equations of the Form- (x-a)²=c
When solving equations of the form: \({\left(x-a\right)}^2=c\), we can use the square root property which states to drop the square and take the positive and negative square root of the constant, c. Mathematically, we say the following:
If \({\left(x-a\right)}^2=c\),
then \(x-a=\pm \sqrt{c}\)
next, add a on both sides of the equation to obtain
\(x=a\pm \sqrt{c}\)
Solve \({\left(x-5\right)}^2=16\)
Solution
To solve, we first drop the square from the left side and take the positive and negative square root of 16 to obtain
\[x-5=\pm \sqrt{16}\]
Next, we simplify the square root
\[x-5=\pm 4\]
Next, we isolate the variable, x by adding 5 on both sides of the equation:
\[x-5+\boldsymbol{5}=+\boldsymbol{5}\pm 4\]
Such that after simplifying, we obtain two answers
\[x=5\pm 4\]
\[x=5+4 \,\,\text{ and }\,\, x=5-4\]
\[x=9\,\, \text{ and }\,\, x=1\]
Solve \(3{\left(x+2\right)}^2+5=80\)
Solution
To solve, we must first isolate the perfect square to create the form: \({\left(x-a\right)}^2=c\), so we first subtract 5 on both sides of the equation, then divide both sides by 3
\[3{\left(x+2\right)}^2+5-\boldsymbol{5}=80-\boldsymbol{5}\]
\[3{\left(x+2\right)}^2=75\]
\[\frac{3\left(x+2\right)^2}{\boldsymbol{3}}=\frac{75}{\boldsymbol{3}}\]
\[{\left(x+2\right)}^2=25\]
Next, we drop the square and take the positive and negative square root of 25 to obtain
\[x+2=\pm \sqrt{25}\]
\[x+2=\pm 5\]
Next, we isolate the x by subtracting 2 on both sides of the equation
\[x+2-\boldsymbol{2}=-\boldsymbol{2}\pm 5\]
After simplifying, we obtain two answers
\[x=-2\pm 5\]
\[x=-2+5 \,\,\text{ and }\,\, x=-2-5\]
\[x=3 \,\,\text{ and }\,\, x=-7\]